Quasitoric manifold

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In mathematics, a quasitoric manifold is a topological analogue of the nonsingular projective toric variety of algebraic geometry. A smooth -dimensional manifold is a quasitoric manifold if it admits a smooth, locally standard action of an -dimensional torus, with orbit space an -dimensional simple convex polytope.

Quasitoric manifolds were introduced in 1991 by M. Davis and T. Januszkiewicz,[1] who called them "toric manifolds". However, the term "quasitoric manifold" was eventually adopted to avoid confusion with the class of compact smooth toric varieties, which are known to algebraic geometers as toric manifolds.[2]

Quasitoric manifolds are studied in a variety of contexts in algebraic topology, such as complex cobordism theory, and the other oriented cohomology theories.[3]

Definitions[edit]

Denote the -th subcircle of the -torus by so that . Then coordinate-wise multiplication of on is called the standard representation.

Given open sets in and in , that are closed under the action of , a -action on is defined to be locally isomorphic to the standard representation if , for all in , in , where is a homeomorphism , and is an automorphism of .

Given a simple convex polytope with facets, a -manifold is a quasitoric manifold over if,

  1. the -action is locally isomorphic to the standard representation,
  2. there is a projection that maps each -dimensional orbit to a point in the interior of an -dimensional face of , for .

The definition implies that the fixed points of under the -action are mapped to the vertices of by , while points where the action is free project to the interior of the polytope.

The dicharacteristic function[edit]

A quasitoric manifold can be described in terms of a dicharacteristic function and an associated dicharacteristic matrix. In this setting it is useful to assume that the facets of are ordered so that the intersection is a vertex of , called the initial vertex.

A dicharacteristic function is a homomorphism , such that if is a codimension- face of , then is a monomorphism on restriction to the subtorus in .

The restriction of λ to the subtorus corresponding to the initial vertex is an isomorphism, and so can be taken to be a basis for the Lie algebra of . The epimorphism of Lie algebras associated to λ may be described as a linear transformation , represented by the dicharacteristic matrix given by

The th column of is a primitive vector in , called the facet vector. As each facet vector is primitive, whenever the facets meet in a vertex, the corresponding columns form a basis of , with determinant equal to . The isotropy subgroup associated to each facet is described by

for some in .

In their original treatment of quasitoric manifolds, Davis and Januskiewicz[1] introduced the notion of a characteristic function that mapped each facet of the polytope to a vector determining the isotropy subgroup of the facet, but this is only defined up to sign. In more recent studies of quasitoric manifolds, this ambiguity has been removed by the introduction of the dicharacteristic function and its insistence that each circle be oriented, forcing a choice of sign for each vector . The notion of the dicharacteristic function was originally introduced V. Buchstaber and N. Ray[4] to enable the study of quasitoric manifolds in complex cobordism theory. This was further refined by introducing the ordering of the facets of the polytope to define the initial vertex, which eventually leads to the above neat representation of the dicharacteristic matrix as , where is the identity matrix and is an submatrix.[5]

Relation to the moment-angle complex[edit]

The kernel of the dicharacteristic function acts freely on the moment angle complex , and so defines a principal -bundle over the resulting quotient space . This quotient space can be viewed as

where pairs , of are identified if and only if and is in the image of on restriction to the subtorus that corresponds to the unique face of containing the point , for some .

It can be shown that any quasitoric manifold over is equivariently diffeomorphic to a quasitoric manifold of the form of the quotient space above.[6]

Examples[edit]

  • The -dimensional complex projective space is a quasitoric manifold over the -simplex . If is embedded in so that the origin is the initial vertex, a dicharacteristic function can be chosen so that the associated dicharacteristic matrix is

The moment angle complex is the -sphere , the kernel is the diagonal subgroup , so the quotient of under the action of is .[7]

  • The Bott manifolds that form the stages in a Bott tower are quasitoric manifolds over -cubes. The -cube is embedded in so that the origin is the initial vertex, and a dicharacteristic function is chosen so that the associated dicharacteristic matrix has given by

for integers .

The moment angle complex is a product of copies of 3-sphere embedded in , the kernel is given by

,

so that the quotient of under the action of is the -th stage of a Bott tower.[8] The integer values are the tensor powers of the line bundles whose product is used in the iterated sphere-bundle construction of the Bott tower.[9]

The cohomology ring of a quasitoric manifold[edit]

Canonical complex line bundles over given by

,

can be associated with each facet of , for , where acts on , by the restriction of to the -th subcircle of embedded in . These bundles are known as the facial bundles associated to the quasitoric manifold. By the definition of , the preimage of a facet is a -dimensional quasitoric facial submanifold over , whose isotropy subgroup is the restriction of on the subcircle of . Restriction of to gives the normal 2-plane bundle of the embedding of in .

Let in denote the first Chern class of . The integral cohomology ring is generated by , for , subject to two sets of relations. The first are the relations generated by the Stanley–Reisner ideal of ; linear relations determined by the dicharacterstic function comprise the second set:

.

Therefore only are required to generate multiplicatively.[1]

Comparison with toric manifolds[edit]

  • Any projective toric manifold is a quasitoric manifold, and in some cases non-projective toric manifolds are also quasitoric manifolds.
  • Not all quasitoric manifolds are toric manifolds. For example, the connected sum can be constructed as a quasitoric manifold, but it is not a toric manifold.[10]

Notes[edit]

  1. ^ a b c M. Davis and T. Januskiewicz, 1991.
  2. ^ V. Buchstaber and T. Panov, 2002.
  3. ^ V. Buchstaber and N. Ray, 2008.
  4. ^ V. Buchstaber and N. Ray, 2001.
  5. ^ V. Buchstaber, T. Panov and N. Ray, 2007.
  6. ^ M. Davis and T. Januskiewicz, 1991, Proposition 1.8.
  7. ^ V. Buchstaber, T. Panov and N. Ray, 2007, Example 3.11.
  8. ^ V. Buchstaber, T. Panov and N. Ray, 2007, Example 3.12.
  9. ^ Y. Civan and N. Ray, 2005.
  10. ^ M. Masuda and D. Y. Suh 2007.

References[edit]

  • Buchstaber, V.; Panov, T. (2002), Torus Actions and their Applications in Topology and Combinatorics, University Lecture Series, 24, American Mathematical Society 
  • Buchstaber, V.; Panov, T.; Ray, N. (2007), "Spaces of polytopes and cobordism of quasitoric manifolds", Moscow Mathematical Journal, 7 (2): 219–242 
  • Buchstaber, V.; Ray, N. (2001), "Tangential structures on toric manifolds and connected sums of polytopes", International Mathematics Research Notices, 4: 193–219 
  • Buchstaber, V.; Ray, N. (2008), "An Invitation to Toric Topology: Vertex Four of a Remarkable Tetrahedron", Proceedings of the International Conference in Toric Topology; Osaka City University 2006, Contemporary Mathematics, 460, American Mathematical Society, pp. 1–27 
  • Civan, Y.; Ray, N. (2005), "Homotopy decompositions and K-theory of Bott towers", K-Theory, 34: 1–33, doi:10.1007/s10977-005-1551-x 
  • Davis, M.; Januskiewicz, T. (1991), "Convex polytopes, Coxeter orbifolds and torus actions", Duke Mathematical Journal, 62 (2): 417–451, doi:10.1215/s0012-7094-91-06217-4 
  • Masuda, M.; Suh, D. Y. (2008), "Classification problems of toric manifolds via topology", Proceedings of the International Conference in Toric Topology; Osaka City University 2006, Contemporary Mathematics, 460, American Mathematical Society, pp. 273–286